This is a task from Kenneth Rosen's "Discrete Mathematics and its applications" book.

Mmmm, Rosen...

Structural induction on full binary trees works as follows. Suppose you have a property P of trees, i.e., for each particular tree T, P(T) is either true or false. Suppose further that you prove that P holds on a single-node tree (consisting of a single root), and for any tree as in the picture above, if and hold, then P of the whole tree holds. In this case, P holds on all full binary trees.

The first step is to come up with P. Here it is easy: P(T) is . Next, prove P for the single-node tree. For the induction step, suppose and hold, i.e.,

and . (*)

Let's call the whole tree T. Express through , and through , . Try to prove P(T) from (*).

Structural induction on full binary trees works as follows. Suppose you have a property P of trees, i.e., for each particular tree T, P(T) is either true or false. Suppose further that you prove that P holds on a single-node tree (consisting of a single root), and for any tree as in the picture above, if and hold, then P of the whole tree holds. In this case, P holds on all full binary trees.

The first step is to come up with P. Here it is easy: P(T) is . Next, prove P for the single-node tree. For the induction step, suppose and hold, i.e.,

and . (*)

Let's call the whole tree T. Express through , and through , . Try to prove P(T) from (*).